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Radon Transform

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The Radon transform is an integral transform whose inverse is used to reconstruct images from medical CT scans. A technique for using Radon transforms to reconstruct a map of a planet's polar regions using a spacecraft in a polar orbit has also been devised (Roulston and Muhleman 1997).

The Radon and inverse Radon transforms are implemented in the Wolfram Language as RadonTransform and InverseRadonTransform, respectively.

The Radon transform can be defined by

R(p,tau)[f(x,y)]=int_(-infty)^inftyf(x,tau+px)dx
(1)
=int_(-infty)^inftyint_(-infty)^inftyf(x,y)delta[y-(tau+px)]dydx
(2)
=U(p,tau),
(3)

where p is the slope of a line, tau is its intercept, and delta(x) is the delta function. The inverse Radon transform is

f(x,y)=1/(2pi)int_(-infty)^inftyd/(dy)H[U(p,y-px)]dp,
(4)

where H is a Hilbert transform. The transform can also be defined by

R^'(r,alpha)[f(x,y)]=int_(-infty)^inftyint_(-infty)^inftyf(x,y)delta(r-xcosalpha-ysinalpha)dxdy,
(5)

where r is the perpendicular distance from a line to the origin and alpha is the angle formed by the distance vector.

Using the identity

F_(omega,alpha)[R[f(omega,alpha)]](x,y)=F_(u,v)^2[f(u,v)](x,y),
(6)

where F is the Fourier transform, gives the inversion formula

f(x,y)=cint_0^piint_(-infty)^inftyF_(omega,alpha)[R[f(omega,alpha)]]|omega|e^(iomega(xcosalpha+ysinalpha))domegadalpha.
(7)

The Fourier transform can be eliminated by writing

f(x,y)=int_0^piint_(-infty)^inftyR[f(r,alpha)]W(r,alpha,x,y)drdalpha,
(8)

where W is a weighting function such as

W(r,alpha,x,y)=h(xcosalpha+ysinalpha-r)
(9)
=F^(-1)[|omega|].
(10)

Nievergelt (1986) uses the inverse formula

f(x,y)=1/pilim_(c->0)int_0^piint_(-infty)^inftyR[f(r+xcosalpha+ysinalpha,alpha)]G_c(r)drdalpha,
(11)

where

G_c(r)={1/(pic^2) for |r|<=c; 1/(pic^2)(1-1/(sqrt(1-c^2/r^2))) for |r|>c.
(12)

Ludwig's inversion formula expresses a function in terms of its Radon transform. R^'(r,alpha) and R(p,tau) are related by

p=cotalpha tau=rcscalpha
(13)
r=tau/(1+p^2) alpha=cot^(-1)p.
(14)

The Radon transform satisfies superposition

R(p,tau)[f_1(x,y)+f_2(x,y)]=U_1(p,tau)+U_2(p,tau),
(15)

linearity

R(p,tau)[af(x,y)]=aU(p,tau),
(16)

scaling

R(p,tau)[f(x/a,y/b)]=|a|U(pa/b,tau/b),
(17)

rotation, with R_phi rotation by angle phi

R(p,tau)[R_phif(x,y)]=1/(|cosphi+psinphi|)U((p-tanphi)/(1+ptanphi),tau/(cosphi+psinphi)),
(18)

and skewing

R(p,tau)[f(ax+by,cx+dy)]=1/(|a+bp|)U[(c+dp)/(a+bp),(tau(ab+bc))/(a+bp)]
(19)

(Durrani and Bisset 1984; correction in Durrani and Bisset 1985).

The line integral along p,tau is

I=sqrt(1+p^2)U(p,tau).
(20)

The analog of the one-dimensional convolution theorem is

R(p,tau)[f(x,y)*g(y)]=U(p,tau)*g(tau),
(21)

the analog of Plancherel's theorem is

int_(-infty)^inftyU(p,tau)dtau=int_(-infty)^inftyint_(-infty)^inftyf(x,y)dxdy,
(22)

and the analog of Parseval's theorem is

int_(-infty)^inftyR(p,tau)[f(x,y)]^2dtau=int_(-infty)^inftyint_(-infty)^inftyf^2(x,y)dxdy.
(23)

If f is a continuous function on C, integrable with respect to a plane Lebesgue measure, and

int_lfds=0
(24)

for every (doubly) infinite line l where s is the length measure, then f must be identically zero. However, if the global integrability condition is removed, this result fails (Zalcman 1982, Goldstein 1993).

See also

Hammer's X-Ray Problems, Inverse Radon Transform, Radon Transform--Cylinder, Radon Transform--Delta Function, Radon Transform--Gaussian, Radon Transform--Square, Tomography

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References

Anger, B. and Portenier, C. Radon Integrals. Boston, MA: Birkhäuser, 1992.Armitage, D. H. and Goldstein, M. "Nonuniqueness for the Radon Transform." Proc. Amer. Math. Soc. 117, 175-178, 1993.Deans, S. R. The Radon Transform and Some of Its Applications. New York: Wiley, 1983.Durrani, T. S. and Bisset, D. "The Radon Transform and its Properties." Geophys. 49, 1180-1187, 1984.Durrani, T. S. and Bisset, D. "Erratum to: The Radon Transform and Its Properties." Geophys. 50, 884-886, 1985.Esser, P. D. (Ed.). Emission Computed Tomography: Current Trends. New York: Society of Nuclear Medicine, 1983.Gindikin, S. (Ed.). Applied Problems of Radon Transform. Providence, RI: Amer. Math. Soc., 1994.Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 1980.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.Helgason, S. The Radon Transform. Boston, MA: Birkhäuser, 1980.Hungerbühler, N. "Singular Filters for the Radon Backprojection." J. Appl. Analysis 5, 17-33, 1998.Kak, A. C. and Slaney, M. Principles of Computerized Tomographic Imaging. IEEE Press, 1988.Kunyansky, L. A. "Generalized and Attenuated Radon Transforms: Restorative Approach to the Numerical Inversion." Inverse Problems 8, 809-819, 1992.Nievergelt, Y. "Elementary Inversion of Radon's Transform." SIAM Rev. 28, 79-84, 1986.Rann, A. G. and Katsevich, A. I. The Radon Transform and Local Tomography. Boca Raton, FL: CRC Press, 1996.Robinson, E. A. "Spectral Approach to Geophysical Inversion Problems by Lorentz, Fourier, and Radon Transforms." Proc. Inst. Electr. Electron. Eng. 70, 1039-1053, 1982.Roulston, M. S. and Muhleman, D. O. "Synthesizing Radar Maps of Polar Regions with a Doppler-Only Method." Appl. Opt. 36, 3912-3919, 1997.Shepp, L. A. and Kruskal, J. B. "Computerized Tomography: The New Medical X-Ray Technology." Amer. Math. Monthly 85, 420-439, 1978.Strichartz, R. S. "Radon Inversion--Variation on a Theme." Amer. Math. Monthly 89, 377-384 and 420-423, 1982.Weisstein, E. W. "Books about Radon Transforms." http://www.ericweisstein.com/encyclopedias/books/RadonTransforms.html.Zalcman, L. "Uniqueness and Nonuniqueness for the Radon Transform." Bull. London Math. Soc. 14, 241-245, 1982.

Referenced on Wolfram|Alpha

Radon Transform

Cite this as:

Weisstein, Eric W. "Radon Transform." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RadonTransform.html

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